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Clark-Okone type martingale representations for Poisson martingales

Martingale representations are fundamental tools of stochastic analysis. In the case of Brownian motion the Clark-Okone representation yields explicit expressions for the integrand in terms of Malliavin derivatives. A similar result is known for Poisson and Lévy processes. In this talk we will explain a general version of this representation for Poisson martingales, taken from [1]. Our first application are short proofs of the Poincare- and the FKG-inequality for Poisson processes. A second application is Wu's [2] elegant proof of a general log-Sobolev inequality for Poisson functionals. If time permits we will also discuss minimal variance hedging or some potential applications in stochastic geometry.